Stability of Discrete Approximations and Necessary Optimality Conditions for Delay-Differential Inclusions

This paper is devoted to the study of optimization problems for dynamical systems governed by constrained delay-differential inclusions with generally nonsmooth and nonconvex data. We provide a variational analysis of the dynamic optimization problems based on their data perturbations that involve finite-difference approximations of time-derivatives matched with the corresponding perturbations of endpoint constraints. The key issue of such an analysis is the justification of an appropriate strong stability of optimal solutions under finite-dimensional discrete approximations. We establish the required pointwise convergence of optimal solutions and obtain necessary optimality conditions for delay-differential inclusions in intrinsic Euler–Lagrange and Hamiltonian forms involving nonconvex-valued subdifferentials and coderivatives of the initial data.

[1]  Thomas S. Angell,et al.  On the necessary conditions for optimal control of retarded systems , 1990 .

[2]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[3]  B. Mordukhovich.  Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions , 1993 .

[4]  J. Warga,et al.  Controllability, extremality, and abnormality in nonsmooth optimal control , 1983 .

[5]  Frank H. Clarke,et al.  Necessary conditions, controllability and the value function for differential-difference inclusions , 1986 .

[6]  I. Ekeland.  On the variational principle , 1974 .

[7]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[8]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[9]  Philip D. Loewen,et al.  Bolza Problems with General Time Constraints , 1997 .

[10]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[11]  Harvey Thomas Banks,et al.  Application of abstract variational theory to hereditary systems--A survey , 1974 .

[12]  Qiji J. Zhu.  Necessary Optimality Conditions for Nonconvex Differential Inclusions with Endpoint Constraints , 1996 .

[13]  B. Mordukhovich.  Discrete Approximations and Refined Euler--Lagrange Conditions forNonconvex Differential Inclusions , 1995 .

[14]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[15]  Peter R. Wolenski,et al.  Necessary conditions for functional differential inclusions , 1996 .

[16]  H. Frankowska.  The maximum principle for an optimal solution to a differential inclusion with end points constraints , 1987 .

[17]  Harry Zheng,et al.  The Extended Euler--Lagrange Condition for Nonconvex Variational Problems , 1997 .

[18]  J. Warga.  Optimal control of differential and functional equations , 1972 .

[19]  Héctor J. Sussmann,et al.  Geometry and optimal control , 1998 .

[20]  Leonid I. Minchenko.  Necessary optimality conditions for differential-difference inclusions , 1999 .

[21]  R. Rockafellar.  Equivalent Subgradient Versions of Hamiltonian and Euler--Llagrange Equations in Variational Analysis , 1996 .

[22]  A. Ioffe.  Euler-Lagrange and Hamiltonian formalisms in dynamic optimization , 1997 .